Well starting with n-of-n Schnorr, the difference is no k^-1 value complicating the math. DSA is a complexification of Schnorr, probably as an attempt to avoid Schnorr's now expired patent. Schnorr is simpler, has better security proofs (possible because its simpler) and makes weaker assumptions about the hash function (ie tolerates a weaker hash, or more slips in hash function properties without breaking the signature, aka DSA stresses the hash function more).
The simplicity makes it easier to do blind signatures, and n of n, k of n etc.
Comparing ECDSA and ECSchnorr (with relabeling to highlight similarities):
ECDSA: R=kG, [r=R.x, s=(H(m)+rd)/k], Q=dG verify: sR=?H(m)G+rQ
ECS: R=kG, [r=R.x, s=k+H(r,m)d], Q=dG verify: sG=?R+H(r,m)Q
ECS-alternate: R=kG, [c=H(R,m), s=k+cd], Q=dG, verify: c=?H(sG-cQ,m)
(because kG=sG-cQ)
(And both ECDSA and ECS can use deterministic variant where k=H(m,d)).
so with ECS if you have users with pub keys A=aG and B=bG (priv keys a,b) they can make a sig with their combined key Q=A+B simply as
R1=k1G, r1=R1.x ->r1
<= r2,s2 R2=k2G, r2=R2.x, s2=k2+H(r1+r2,m)b
s1=k1+H(r1+r2,m)a, r=r1+r2, s=s1+s2
as r1+r2=k1G+k2G=(k1+k2)G, s1+s2=(k1+k2)+H(r1+r2,m)(a+b)
As there was discussion on this topic on twitter I thought I'd update this with how to bootstrap from 2 of 2 to 2 of 3: introduce new signer C
to recap the combined public key Q=A+B
combined private key d=a+b
we will re-split d twice, once by A and once by b:
first A:
1. A choses random r and sets a'=a-r
2. A sends r to B
3. B sets b'=b+r
4. A sends a' to C
(as d=a'+b'=a-r+b+r=a+b, so a',b' is a re-split of d)
similarly B:
5. B choses random r' and sets b"=b-r'
6. B sends r' to A
7. A sets a"=a+r'
8. B sends b" to C
(as d=a"+b"=a-r'+b+r'=a+b, so a",b" is a second re-split of d)
Now any 2 of A,B or C can sign consider the three cases:
A & B: sign with a,b
A & C sign with a",b" (as B sent b" to C, this prevents A signing by itself)
B & C sign with a',b' (as A sent a' to C, this prevents B signing by itself).
This setup pattern scales to other k of n thresholds.
Adam